$A_n^{(1)}$-Geometric Crystal corresponding to Dynkin index $i=2$ and its ultra-discretization

Abstract

Let $g$ be an affine Lie algebra with index set $I=\{0,1,2,...,n\}$ and $g^L$ be its Langlands dual. It is conjectured that for each $i\in I\setminus \{0\}$ the affine Lie algebra $g$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for $g^L$. We prove this conjecture for $i=2$ and $g=A_n^{(1)}$.

arXiv comment18 pages. arXiv admin note: substantial text overlap with arXiv:1003.1242, arXiv:0712.3894, arXiv:math/0612858, arXiv:0911.3556